CK-12 Geometry Concepts

9.7. Angles On and Inside a Circle http://www.ck12.org

1. Draw

⊙

Awith chordBCand tangent line

←→

EDwith point of tangencyC.

2. Draw in central angle^6 CAB. Then, using your protractor, findm^6 CABandm^6 BCE.


3. FindmBĈ(the minor arc). How does the measure of this arc relate tom^6 BCE?

This investigation proves the Chord/Tangent Angle Theorem.

Chord/Tangent Angle Theorem:The measure of an angle formed by a chord and a tangent that intersect on the
circle is half the measure of the intercepted arc.

From the Chord/Tangent Angle Theorem, we now know that there are two types of angles that are half the measure
of the intercepted arc; an inscribed angle and an angle formed by a chord and a tangent. Therefore,any angle with
its vertex on a circle will be half the measure of the intercepted arc.

An angle is considered inside

Investigation: Find the Measure of an Angle

Tools Needed: pencil, paper, compass, ruler, protractor, colored pencils (optional)

1. Draw
   ⊙
   Awith chordBCandDE. Label the point of intersectionP.


2. Draw central angles^6 DABand^6 CAE. Use colored pencils, if desired.


3. Using your protractor, findm^6 DPB,m^6 DAB, andm^6 CAE. What ismDB̂andmCÊ?


4. FindmDB̂+ 2 mCÊ.


5. What do you notice?

Intersecting Chords Angle Theorem:The measure of the angle formed by two chords that intersectinsidea circle
is the average of the measure of the intercepted arcs.

Inthepicturebelow:

m^6 SV R=

1

2

(

m̂SR+mT Q̂

)

=

mSR̂+mT Q̂

2

=m^6 T V Q

m^6 SV T=

1

2

(

mST̂+mRQ̂

)

=

mST̂+mRQ̂

2

=m^6 RV Q

Example A

FindmAEB̂

Use the Chord/Tangent Angle Theorem.

mAEB̂= 2 ·m^6 DAB= 2 · 133 ◦= 266 ◦

Example B

Findm^6 BAD.

Use the Chord/Tangent Angle Theorem.

m^6 BAD=^12 mAB̂=^12 · 124 ◦= 62 ◦

Example C

Finda,b, andc.